Assuming that in the ~584 odd years from the study to the present day that 20 generations passed (average of 30 years per generation), the specific great-great-grandparent who passed on that name has roughly a 0.5^30 on that person (which is an infinitesimally small number).
However, rich people usually marry rich people. It's likely these results aren't just direct inheritance, or direct genetics, but it shows stratification within Italian society.
You are right that something's been passed down - but it's not the genes. It's the social connection to:
* Get jobs without applying for them.
* To learn important economic news before anyone hears them. E.g. President bought 1200 Cuban cigar just before banning imports from CUBA. Read this: https://www.cigaraficionado.com/index.php/article/great-mome...
* To get contracts regardless of ones qualifications. I've read that Aristocrats ensure their kids have ivy league degrees to maintain a illusion of fairness.
There's an article I read here on HN that says, Ivy league schools are successful because they select applicants who would be successful without them.
See e.g. https://helda.helsinki.fi/bitstream/handle/10138/38881/HECER...
That sounds like a claim that there's no causal connection running through genes.
Propensity to lying, cheating, saving, nepotism, length of life, fecundity, access to healthcare, intelligence, math ability and so many more factors are easily inheritable.